NANO
LETTERS
Experimental Observation of Nonlinear
Ionic Transport at the Nanometer Scale
2006
Vol. 6, No. 11
2531-2535
Diego Krapf, Bernadette M. Quinn, Meng-Yue Wu, Henny W. Zandbergen,
Cees Dekker, and Serge G. Lemay*
KaVli Institute of Nanoscience, Delft UniVersity of Technology, Lorentzweg 1,
2628 CJ Delft, The Netherlands
Received August 18, 2006; Revised Manuscript Received September 25, 2006
ABSTRACT
Nanometer-sized electrodes are used to probe the transport of ions in liquid by monitoring heterogeneous electrochemical reactions. We
observe pronounced nonlinearities of ion flux versus concentration when transport is localized within a region smaller than 10 nm. We show
that these observations cannot be explained using conventional continuum, mean-field descriptions of ionic transport. The data indicate that
these deviations are caused by the high flux of charged species that is achieved at nanometer-sized electrodes.
Understanding ionic transport in liquid near charged interfaces is an outstanding theoretical and experimental challenge
of particular relevance to the emerging field of nanofluidics.
The equilibrium distributions and transport properties of ionic
systems are most commonly described using the PoissonBoltzmann (PB) and Poisson-Nernst-Planck (PNP) formalisms, respectively. Such continuum, mean-field descriptions
can break down on sufficiently small length scales because
they take into account neither the discreteness and finite size
of the mobile ions nor the structure of the solvent. Experiments that probe these microscopic length scales directly are
now becoming possible. For example, recent X-ray scattering
measurements of the equilibrium ion distribution near
charged interfaces1,2 and DNA molecules3 required taking
into account the effect of local chemical equilibrium,1 finite
ion sizes,3 or molecular-scale solvent structure.2
Compared to equilibrium properties, the transport of ions
near charged interfaces remains poorly understood. The
difficulty lies in disentangling the contributions from electrostatics, hydrodynamics, and solvent structure on the scale
of a few nanometers. Several recent experimental studies
have concentrated on ionic transport parallel to charged
interfaces, which can be probed in systems ranging from
colloids4 to synthetic5,6 or biological7,8 nanochannels. Transport perpendicular to an interface, however, couples more
directly to the electrostatic potential induced by surface
charges. A powerful method for measuring conductance
across the double layer is electrochemistry, where charge
transfer between redox molecules in solution and an electrode
provides a probe of ionic transport. Although the NernstPlanck equation with the electroneutrality approximation was
* Corresponding
author.
lemay@mb.tn.tudelft.nl.
10.1021/nl0619453 CCC: $33.50
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Figure 1. (A) Schematic representation of the experimental setup.
A nanoelectrode is immersed in solution, a potential is applied,
and the current is recorded. (B) Transmission electron microscopy
image of a pore with radius 2.5 nm in a SiN membrane. (C)
Voltammetric response of an electrode 5 nm in radius (0.1 mM
FcTMA+ in 0.5 M NH4NO3).
initially applied to this problem,9,10 the advent of nanometerscale electrodes11 revived the question of the validity of this
simple approach.12,13 More recently, anomalous transport to
submicrometer electrodes was observed upon removal of
supporting electrolyte,14-16 which was attributed to the
interfacial electric field.
Here we study ion transport using gold electrodes with
radii of only a few nanometers. This size approaches both
the typical interionic spacing and the Debye screening length,
even in high ionic strength solutions. As illustrated in Figure
1A, a nanoelectrode is immersed in an electrolyte and biased
so as to drive an electron-transfer reaction with an ionic
species in solution. The electrical current through the
electrode provides a direct measure of the flux of ions
transported to the electrode surface. Because the concentration gradient is localized within a length scale comparable
to the dimensions of the nanoelectrode, extremely steep
stationary concentration gradients are achieved and transport
properties are probed on the scale of several nanometers.
As a result of this abrupt concentration gradient, exceptionally high ionic fluxes are reached. We find that the classical
mean-field approximation does not describe ionic transport
adequately in this regime. Specifically, the data indicate that
the PNP formalism fails because of the nonequilibrium
conditions induced at the electrical double layer.
To carry out these experiments, we developed a method
for fabricating nanoelectrodes with a well-defined geometry
and independently determined dimensions. The latter is
particularly important to the present work: in previous
studies, an effective radius was typically deduced from the
steady-state electrochemical current itself without the possibility of independently characterizing electrodes smaller
than ∼50 nm.11-16 We show below that the current no longer
scales linearly with electrode size for radii below ∼10 nm,
invalidating this procedure.
The details of the electrode fabrication process were
reported previously.17 In brief, we first fabricated a 20-nmthick free-standing SiN membrane using standard microfabrication technology. Pores with radii between 2 and 150 nm
were then drilled in the thin membranes with a focused
electron beam.18,19 Following pore formation, high-resolution
transmission electron microscopy was used to determine the
radius of each individual pore with subnanometer accuracy
(Figure 1B). The pores were subsequently filled with gold
to yield convex electrodes with a diameter equal to that of
the pore, as sketched in Figure 1A.
Ferrocenylmethyltrimethylammonium (FcTMA+) and ferrocenedimethanol (Fc(CH2OH)2) were employed as electroactive species. FcTMA+ is oxidized at the electrode surface
to become FcTMA++ at a formal potential of 0.42 V with
respect to a Ag/AgCl reference. Fc(CH2OH)2 is neutral and
is oxidized to Fc(CH2OH)2+ at 0.25 V. For the electrochemical measurements, a standard 3 M Ag/AgCl electrode served
as both reference- and counter-electrode. The oxidation of
the redox species was recorded using home-built electronics.
Our preamplifier had a 3 Hz bandwidth and 2 fA rms noise.
Each data point was integrated over a period of 1 s following
a 1 s delay, ensuring that we measured the steady-state
current. Ion concentrations were deduced from the diffusionlimited current at a commercial 10 µm disc electrode.20
Excess electroinactive NH4NO3 (0.5 M) was added to the
solutions as base electrolyte (Debye length ≈ 0.4 nm).
Electrochemical currents are governed by both the transport
of reactants to the electrode and the electron-transfer rate at
the solution-electrode interface. The latter varies exponentially with the electrode potential.21 Because we are interested
in probing ionic transport, we concentrate on potentials where
charge-transfer kinetics do not play any role. This situation
is always achieved at sufficiently high potentials, where
2532
Figure 2. (A) Transport-limited current as a function of FcTMA+
concentration and Fc(CH2HO)2 at an electrode 19 nm in radius and
(B) at an electrode 7 nm in radius. The dashed lines show the
expected behavior (no fitting parameters). (C) Transport-limited
current as a function of concentration of FcTMA+ in electrodes
with radii of 3.5, 5, and 7 nm. Solid lines are guides to the eye,
and dashed lines show the expected diffusion-driven current based
on the known electrode size.
electron-transfer is extremely fast (i.e., V - V0 . KT/e, where
V is the applied potential and V0 is the formal potential of
the reaction).
A typical cyclic voltammogram (current-voltage response) is shown in Figure 1C. The current reaches a plateau,
indicating that it is independent of heterogeneous kinetics
and is controlled by ionic transport at potentials higher than
0.5 V. In addition to the faradaic current, a finite slope and
a hysteretic offset are observed between the forward and
backward scans because of the parasitic dielectric response
of the insulating membrane.17 Two methods were used to
subtract these dielectric currents: either voltammograms of
solutions containing no electroactive ions were recorded and
subtracted from the original data or the electrode was set at
the plateau potential and the current was measured until
stable. Both methods yielded the same results.
Transport-limited currents (plateaus in the voltammograms) at an electrode 19 nm in radius for different ionic
concentrations are shown in Figure 2A. The currents, and
therefore the ionic fluxes, vary linearly with concentration.
This is consistent with the widely used assumption that in
the presence of excess supporting electrolyte, mass-transport
is dominated by diffusion of the redox species.10,21 At a
hemispherical electrode, this yields iD ) 2πenDR0, with n
being the ion number density, D being the diffusion
coefficient, and R0 being the electrode radius. The dashed
lines in the figure show the predicted behavior approximating
the geometry of our electrodes to a hemisphere using the
measured radius and known diffusion constants. These curves
match the data quite well without any adjustable parameters.
Nano Lett., Vol. 6, No. 11, 2006
We now demonstrate that this effect cannot be captured
by the PNP formalism, which has been used (in either full
or simplified forms) to describe practically all electron
transfer measurements at nanoelectrodes to date.9-16,22 In the
absence of convective flows, the PNP equations are
∇2φ ) -eΣ zi ni
(
B ni +
B
J i ) -Di ∇
Figure 3. (A) Transport-limited current of FcTMA+ normalized
to the calculated diffusion-limited current as a function of electrode
radius for eight electrodes and three different ion concentrations:
0.25, 0.5, and 1 mM. The lines are solutions of the PNP equations
with dOHP ) 0.3, 0.3, and 0.18 nm and dPET ) 0.4, 0.3, and 0.18
nm (solid, dashed, and dotted lines respectively). (B) Potential at
a 5 nm electrode calculated using PNP. The outer Helmholtz plane
(1.3 nm) and the plane of electron-transfer (1.4 nm) are shown.
(C) Simulation results showing a comparison bewteen FcTMA+
concentration obtained using PNP and pure diffusion.
Strong deviations from the simple diffusive behavior are
observed for electrodes with sub-10-nm dimensions. As
shown in Figure 2C, transport to these electrodes is no longer
linear with FcTMA+ concentration, instead increasing in a
sublinear manner. The predicted diffusion-driven currents are
also shown in the Figure, showing that the current is
suppressed when the FcTMA+ concentration exceeds ∼0.2
mM. This dramatic departure from classical diffusion occurs
for the charged FcTMA+ but was not observed for electroneutral Fc(CH2OH)2, as shown in Figure 2B. This behavior
indicates that the current suppression is induced by coupling
of the charged species to the electric field.
Figure 3A shows the limiting currents at 8 electrodes with
radii in the 2-20 nm range using different FcTMA+
concentrations. The electrode potential and base electrolyte
concentration for all data presented here were 0.5 V and 0.5
M, respectively. To compare electrodes of different sizes,
the measured currents were normalized to the calculated
diffusion-driven current, iD, based on the known electrode
radius. The measured currents are systematically smaller than
the expected diffusion-driven values (i/iD < 1), and the
degree of suppression is greater the higher the FcTMA+
concentration and the smaller the electrode size.
The failure of pure diffusion to account for our observations with charged redox molecules is expected because the
size of our electrodes starts to approach the values of both
the Debye length and the size of the ions. The interfacial
electric field should therefore affect the transport of charged
molecules. Nonetheless, because the base electrolyte concentration was kept constant and much greater than the
FcTMA+ concentration, the electric field is expected to be
independent of electroactive species concentration. A strong
nonlinear dependence on electroactive ion concentration,
such as that seen in Figure 2C, is therefore extremely
surprising.
Nano Lett., Vol. 6, No. 11, 2006
zi e
Bφ
n∇
kT i
(1)
)
(2)
where φ is the electrostatic potential, is the electric
permittivity, B
Ji and zi are the flux and valence of ion species
i, respectively, and the rest of the symbols were defined
above. Subindex i spans the four species involved in the
problem: two for the base electrolyte (z ) (1) and two for
the electroactive molecules (z ) +1; +2 for FcTMA). In
steady state, mass conservation implies ∇
B‚Ji ) 0 for the redox
couple and B
Ji ) 0 for the inert ions. We employ standard
boundary conditions for the electrode surface. Namely, the
electrode is treated in terms of an outer Helmholtz plane
(OHP) and a plane of electron transfer (PET, Figure 3B).
The OHP serves as the plane of closest approach of ions to
the electrode surface. The region between the electrode
surface proper and the OHP has zero charge density, and
continuity is maintained for φ and ∇
B φ at the OHP. We allow
electron transfer to occur further away than the OHP, at a
distance dPET from the surface. At the PET, the reactant
concentration is zero (fast electron-transfer approximation)
and the product flux equals the reactant flux with opposite
sign (mass conservation).
The equations were solved numerically using a finite
elements package (Comsol Multiphysics). We considered a
hemispherical electrode geometry so that the problem simplified to spherical symmetry. The resulting potential and
FcTMA+ concentration profiles for a 5 nm electrode are
plotted in Figure 3B and 3C. The electrode potential is not
uniquely defined because the potential of zero charge (pzc)
of the system is not known. However, the pzc of gold versus
Ag/AgCl is on the order of 0.04 V.15 We checked that the
simulations are not sensitive to changes in the electrode
voltage within the relevant range of 0.4-0.6 V and that our
results are not qualitatively changed by this potential.
Although the distances dOHP and dPET can be estimated, their
exact values cannot be calculated and are generally determined empirically. We therefore performed calculations over
a broad range of parameters. Typical results are represented
in Figure 3A. Suppression of the current is indeed predicted
with decreasing electrode size below 50 nm, which has its
physical origin in the overlap between the diffusion region
and the electrical double layer.13-15,22 Results are essentially
insensitive to the position of the OHP (for physically relevant
values of 0.2-0.4 nm). The current is, however, very
sensitive to dPET. In particular, the degree of suppression
increases with decreasing dPET, consistent with the rapid
decay of the (equilibrium) electrostatic potential with distance
from the electrode.
Most relevant to our experimental observations, PNP
predicts that the normalized current, i/iD, is insensitive
2533
Figure 4. Normalized transport-limited current as a function of
current density for electrodes with radius between 3.5 and 19 nm.
The three lines show the PNP solutions with OHP at 0.3 nm and
charge transfer at 0.4 nm for electrode radii of 19, 5, and 3.5 nm.
(difference invisible on the scale of Figure 3A) to the
concentration of electroactive ions within the range of
concentrations probed, contradicting the data in Figures 2C
and 3A. Although Figure 3A shows that individual data sets
at particular concentrations can be fitted adequately by
selecting appropriate values for dPET, the parameters obtained
are not physical. Data at different concentrations require
different values of dPET and are impossible to fit with a single
choice of this parameter. Because dOHP and dPET represent
local interactions between individual ions and the electrode,
they cannot depend on the electroactive ion concentration.
We conclude that the conventional PNP model fails to
explain the pronounced nonlinearity of the electrochemical
current with concentration.
Suppression of the current is observed for electrode radii
comparable to or smaller than the average spacing between
electroactive ions, R0 ≈ n-1/3 ≈ 10-20 nm. Figures 2C and
3A, however, show that for a given electrode size the degree
of suppression increases with increasing ion concentration.
This indicates that this crossover in length scales is not
responsible for the current suppression.
Some understanding of this surprising effect is gained by
focusing on the nonequilibrium conditions at the electrode
surface. Figure 4 shows the observed degree of suppression,
i/iD, versus the measured current density at the electrode
i/2πR02. The latter is a measure of how far the system is
driven from equilibrium. Data for different electrodes collapse within error unto a single curve. At low current
densities, i/iD approaches unity and pure diffusion provides
an adequate estimate of the measured current. At current
densities less than ∼0.15 A/cm2, the current is suppressed
below the expected value for pure diffusion but still lies
within experimental error of the values expected from the
PNP calculations. At current densities above ∼0.15 A/cm2,
significant additional suppression is observed, which cannot
be described by PNP.
This analysis implies that the simple PNP model breaks
down in the vicinity of the electrode when the system is
2534
driven sufficiently far from equilibrium. Deviations are thus
observed at nanoelectrodes because much higher flux densities can be achieved rather than because of a crossover in
length scales. Therefore, the observed critical radius is set
by the electrode size at which sufficiently high current
densities are attained for practical ionic concentrations (i.e.,
within the solubility limit). Limits to the validity of the meanfield approximation are relatively well-understood for equilibrium distributions of ions, but this is not true for dynamical
situations, especially for high current densities within small
length scales that were previously experimentally inaccessible. Recently, several authors have questioned the applicability of transport continuum models to nanoscale
systems.23-25 Ion permeation through biological channels was
simulated using all-atom molecular dynamics (MD)26,27 as
well as Brownian dynamics.23,24 It was concluded that the
PNP formalism does not capture the physics of small
volumes involving the statistics of few ions. This analysis
has not yet been brought to bear on electrochemical
problems, however. Here, additional mechanisms neglected
in PNP are likely to become significant far from equilibrium,
including hydrodynamic coupling, which appears to not play
an important role in ionic channels. Furthermore, the
diffusive transport of individual ions is coupled via collective
effects such as counterion atmosphere relaxation and associated density fluctuations, which can be described using, for
example, mode coupling theory.28 Disentangling these effects
will, however, require extensive theoretical work.
In summary, we have performed reliable measurement of
ionic transport at independently characterized nanoelectrodes.
We observed a pronounced deviation from predictions based
on continuum models for electrodes smaller than ∼10 nm.
This behavior is governed by the ion flux at the electrode
surface. This demonstrates the need for more advanced
models to describe far-from-equilibrium ionic transport when
high ion fluxes are attained locally in nanometer domains.
Acknowledgment. This work was funded by NanoNed,
FOM, and NWO.
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